The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?
Prove that in a game of noughts and crosses on a \(3\times 3\) grid, if the first player uses the right strategy then the second player cannot win.
There are some coins on a table. One of these coins is fake (has a different weight than a real coin). By weighing them twice using balance scales, determine whether the fake coin is lighter or heavier than a real coin (you don’t need to find the fake coin) if the number of coins is: a) 100; b) 99; c) 98?
Can you find
a) in the 100th line of Pascal’s triangle, the number \(1 + 2 + 3 + \dots + 98 + 99\)?
b) in the 200th line the sum of the squares of the numbers in the 100th line?
On a function \(f (x)\), defined on the entire real line, it is known that for any \(a>1\) the function \(f (x) + f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.
On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?
Prove that the infinite decimal \(0.1234567891011121314 \dots\) (after the decimal point, all of the natural numbers are written out in order) is an irrational number.
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.